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Abstract Well-known debates among statistical inferential paradigms emerge from conflicting views on the notion of probability. One dominant view understands probability as a representation of sampling variability; another prominent view understands probability as a measure of belief. The former generally describes model parameters as fixed values, in contrast to the latter. We propose that there are actually two versions of a parameter within both paradigms: a fixed unknown value that generated the data and a random version to describe the uncertainty in estimating the unknown value. An inferential approach based on CDs deciphers seemingly conflicting perspectives on parameters and probabilities. 

Approximate confidence distribution computing (ACDC) offers a new take on the rapidly developing field of likelihood-free inference from within a frequentist framework. The appeal of this computational method for statistical inference hinges upon the concept of a confidence distribution, a special type of estimator which is defined with respect to the repeated sampling principle. An ACDC method provides frequentist validation for computational inference in problems with unknown or intractable likelihoods. The main theoretical contribution of this work is the identification of a matching condition necessary for frequentist validity of inference from this method. In addition to providing an example of how a modern understanding of confidence distribution theory can be used to connect Bayesian and frequentist inferential paradigms, we present a case to expand the current scope of so-called approximate Bayesian inference to include non-Bayesian inference by targeting a confidence distribution rather than a posterior. The main practical contribution of this work is the development of a data-driven approach to drive ACDC in both Bayesian or frequentist contexts. The ACDC algorithm is data-driven by the selection of a data-dependent proposal function, the structure of which is quite general and adaptable to many settings. We explore three numerical examples that both verify the theoretical arguments in the development of ACDC and suggest instances in which ACDC outperform approximate Bayesian computing methods computationally.